Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity-II.

This paper deals with the following mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity u t − Δ u t − d i v (| ∇ u | p − 2 ∇ u) = | u | q − 2 u log ⁡ | u | in a bounded domain with zero Dirichlet boundary condition, which was studied in our previous paper [J Math Anal Appl. 2019;478(2):393-420]. In view the results of [J Math Anal Appl. 2019;478(2):393-420], for the case (1) 1 < p ≤ q ≤ 2 , i f n ≤ p , ≤ 2 , i f 2 n n + 2 < p < n , < n p n − p , i f p ≤ 2 n n + 2 , the global existence and blow-up results were got when J (u 0) ≤ d , where d denotes the mountain-pass level. But for the case (2) 1 < p ≤ q a n d 2 < q < ∞ , i f n ≤ p , n p n − p , i f 2 n n + 2 < p < n , the blow-up results were got when J (u 0) ≤ M , where M ≤ d is a constant. In this paper, we extend and complete the results of [J Math Anal Appl. 2019;478(2):393-420] on the following three aspects: First, the blow-up results are got when J (u 0) ≤ d and (2) are satisfied. Second, the upper and lower bounds of blow-up time are estimated. Third, the global existence and blow-up results are got when J (u 0) > d. [ABSTRACT FROM AUTHOR]